Systemic juvenile idiopathic arthritis (SJIA) is type of arthritis that typically occurs before 16 years of age. SJIA usually involves heat, pain, swelling, and stiffness in the body's joints. It can also involve fever, rash, anemia, and inflammation in various parts of the body. Rilonacept is a drug that can reduce inflammation. The purpose of this study is to determine whether a rilonacept drug regimen initiated early is more effective than a similar rilonacept drug regimen initiated 3 weeks later when treating children and young adults with SJIA. The current standard treatment for SJIA includes nonsteroidal anti-inflammatory drugs (NSAIDS) and corticosteroids. However, in most people, NSAIDS do not completely control the disease. Also, no studies have been done to prove which medication or combination of medications is best to treat children and adolescents with SJIA. Interleukin-1 (IL-1), a protein secreted by certain cells in the body, assists in regulating immune and inflammatory responses. Too much IL-1 can be harmful and has been shown to play a role in the inflammation associated with a variety of diseases, including SJIA. Rilonacept is a drug that inhibits IL-1 activity. The purpose of this study is to determine whether a rilonacept drug regimen initiated early is more effective than a similar rilonacept drug regimen initiated 3 weeks later when treating children and young adults with SJIA. This study will also evaluate the safety of rilonacept, and various tissue samples will be collected from participants for future genetic studies.